Thermodynamics in the Limit of Irreversible Reactions
A. N. Gorban
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK
arXiv:1207.2507v2 [cond-mat.stat-mech] 11 Oct 2012
E. M. Mirkes
Institute of Space and Information Technologies, Siberian Federal University, Krasnoyarsk, Russia
G. S. Yablonsky
Parks College, Department of Chemistry, Saint Louis University, Saint Louis, MO 63103, USA
Abstract
For many real physico-chemical complex systems detailed mechanism includes both reversible and irreversible
reactions. Such systems are typical in homogeneous combustion and heterogeneous catalytic oxidation.
Most complex enzyme reactions include irreversible steps. The classical thermodynamics has no limit for
irreversible reactions whereas the kinetic equations may have such a limit. We represent the systems with
irreversible reactions as the limits of the fully reversible systems when some of the equilibrium concentrations
tend to zero. The structure of the limit reaction system crucially depends on the relative rates of this
tendency to zero. We study the dynamics of the limit system and describe its limit behavior as t → ∞. If
the reversible systems obey the principle of detailed balance then the limit system with some irreversible
reactions must satisfy the extended principle of detailed balance. It is formulated and proven in the form of
two conditions: (i) the reversible part satisfies the principle of detailed balance and (ii) the convex hull of
the stoichiometric vectors of the irreversible reactions does not intersect the linear span of the stoichiometric
vectors of the reversible reactions. These conditions imply the existence of the global Lyapunov functionals
and alow an algebraic description of the limit behavior. The thermodynamic theory of the irreversible limit
of reversible reactions is illustrated by the analysis of hydrogen combustion.
Keywords: entropy, free energy, reaction network, detailed balance, irreversibility
PACS: 05.45.-a, 82.40.Qt, 82.20.-w, 82.60.Hc
1. Introduction
1.1. The problem: non-existence of thermodynamic functions in the limit of irreversible reactions
We consider a homogeneous chemical system with n components Ai , the concentration of Ai is ci ≥ 0,
the amount of Ai in the system is Ni ≥ 0, V is the volume, Ni = V ci , T is the temperature. The n
dimensional vectors c = (ci ) and N = (Ni ) belong to the closed positive orthant Rn+ in Rn . (Rn+ is the set
of all vectors x ∈ Rn such that xi ≥ 0 for all i.)
The classical thermodynamics has no limit for irreversible reactions whereas the kinetic equations have.
For example, let us consider a simple cycle
k1
k2
k3
k−1
k−2
k−3
A1 ⇋ A2 ⇋ A3 ⇋ A1
Email address: [email protected] (A. N. Gorban)
Preprint submitted to Elsevier
October 12, 2012
eq eq
with the equilibrium concentrations ceq = (ceq
1 , c2 , c3 ) and the detailed balance conditions:
eq
ki ceq
i = k−i ci+1
under the standard cyclic convention, here, A3+1 = A1 and c3+1 = c1 . The perfect free energy has the form
X
ci
RT V ci ln eq − 1 + const .
F =
ci
i
eq
Let the equilibrium concentration ceq
1 → 0 for the fixed values of c2,3 > 0. This means that
ceq
k3
ceq
k−1
1
= 1eq → 0 and
= eq
→ 0.
k1
c2
k−3
c3
Let us take the fixed values of the rate constants k1 , k±2 and k−3 . Then the limit kinetic system exists and
has the form:
k2
k1
A1 →A
2 ⇋ A3 ← A1 .
k−2
k−3
It is a routine task to write a first order kinetic equation for this scheme. At the same time, the free energy
function F has no limit: it tends to ∞ for any positive vector of concentrations because the term c1 ln(c1 /ceq
1 )
increases to ∞. The free energy cannot be normalized by adding a constant term because the variation of
eq
the term c1 ln(c1 /ceq
1 ) on an interval [0, c] with fixed c also increases to ∞, it varies from −c1 /e (for the
eq
eq
minimizer, c1 = c1 /e) to a large number c(ln c − ln c1 ) (for c1 = c).
The logarithmic singularity is rather “soft” and does not cause a real physical problem because even for
−10
ceq
/c
the corresponding large term in the free energy will be just ∼ 23RT per mole. Nevertheless,
1 = 10
1
the absence of the limit causes some mathematical questions. For example, the density,
X
ci (ln(ci /ceq
(1)
f = F/(RT V ) =
i ) − 1) ,
i
is a Lyapunov function for a system of chemical kinetics for a perfect mixture with detailed balance under
isochoric isothermal conditions. Here, ci is the concentration of the ith component and ceq
i is its equilibrium
concentration for a selected value of the linear conservation laws, the so-called “reference equilibrium”.
This function is used for analysis of stability, existence and uniqueness of chemical equilibrium since
the work of Zeldovich (1938, reprinted in 1996 [26]). Detailed analysis of the connections between detailed
balance and the free energy function was provided in [19]. Perhaps, the first detailed proof that f is a
Lyapunov function for chemical kinetics of perfect systems with detailed balance was published in 1975 [22].
Of course, it does not differ significantly from the Boltzman’s proof of his H-theorem (1873 [2]).
For the irreversible systems which are obtained as limits of the systems with detailed balance, we should
expect the preservation of stability of the equilibrium. Moreover, one can expect existence of the Lyapunov
functions which are as universal as the thermodynamic functions are. The “universality” means that these
functions depend on the list of components and on the equilibrium concentrations but do not depend on the
reaction rate constants directly.
The thermodynamic potential of a component Ai cannot be defined in the irreversible limits when
the equilibrium concentration of Ai tends to 0. Nevertheless, in this paper, we construct the universal
Lyapunov functions for systems with some irreversible reactions. Instead of detailed balance we use the
weaker assumption that these systems can be obtained from the systems with detailed balance when some
constants tend to zero.
1.2. The extended form of detailed balance conditions for systems with irreversible reactions
Let us consider a reaction mechanism in the form of the system of stoichiometric equations
X
X
βrj Aj (r = 1, . . . , m) ,
αri Ai →
i
j
2
(2)
where αri ≥ 0, βrj ≥ 0 are the stoichiometric coefficients. The reverse reactions with positive rate constants
are included in the list (2) separately (if they exist). The stoichiometric vector γr of the elementary reaction
is γr = (γri ), γri = βri − αP
ri . We always assume that there exists a strictly positive conservation law, a
vector b = (bi ), bi > 0 and i bi γri = 0 for all r. This may be the conservation of mass or of total number
of atoms, for example.
According to the generalized mass action law, the reaction rate for an elementary reaction (2) is (compare
to Eqs. (4), (7), and (14) in [14] and Eq. (4.10) in [7])
n
Y
wr = kr
ri
,
aα
i
(3)
i=1
where ai ≥ 0 is the activity of Ai ,
ai = exp
µi − µ0i
RT
.
(4)
Here, µi is the chemical potential and µ0i is the standard chemical potential of the component Ai .
This law has a long history (see [6, 24, 13, 7]). It was invented in order to meet the thermodynamic
restrictions on kinetics. For this purposes, according to the principle of detailed balance, the rate of the
reverse reaction is defined by the same formula and its rate constant should be found from the detailed
balance condition at a given equilibrium.
It is worth mentioning that the free energy has no limit when some of the reaction equilibrium constants
tend to zero. For example, for the ideal gas the chemical potential is µi (c, T ) = RT ln ci + µ0i (T ). In the
irreversible limit some µ0i → ∞. On the contrary, the activities remain finite (for the ideal gases ai = ci )
and the approach based on the generalized mass action law and the detailed balance equations wr+ = wr−
can be applied to find the irreversible limit.
The list (2) includes reactions with the reaction rate constants kr > 0. For each r we define kr+ = kr ,
+
wr = wr , kr− is the reaction rate constant for the reverse reaction if it is on the list (2) and 0 if it is not, wr−
is the reaction rate for the reverse reaction if it is on the list (2) and 0 if it is not. For a reversible reaction,
Kr = kr+ /kr−
The principle of detailed balance for the generalized mass action law is: For given values kr there exists
+
−
a positive equilibrium aeq
i > 0 with detailed balance, wr = wr .
Recently, we have formulated the extended form of the detailed balance conditions for the systems with
some irreversible reactions [12]. This extended principle of detailed balance is valid for all systems which
obey the generalized mass action law and are the limits of the systems with detailed balance when some of
the reaction rate constants tend to zero. It consists of two parts:
• The algebraic condition: The principle of detailed balance is valid for the reversible part. (This means
that for the set of all reversible reactions there exists a positive equilibrium where all the elementary
reactions are equilibrated by their reverse reactions.)
• The structural condition: The convex hull of the stoichiometric vectors of the irreversible reactions
has empty intersection with the linear span of the stoichiometric vectors of the reversible reactions.
(Physically, this means that the irreversible reactions cannot be included in oriented cyclic pathways.)
Let us recall the formal convention: the linear span of empty set is {0}, the convex hull of empty set is
empty.
In our previous work [12] we studied the systems with some irreversible reactions which are the limits
of the reversible systems with detailed balance. The structural and algebraic conditions were found. The
present paper is focused on the dynamical consequences of these conditions. We prove that the attractors
always consist of the fixed points. These limit points (“partial equilibria”) are situated on the faces of the
positive orthant of concentrations. These faces and the partial equilibria are described in the paper.
3
1.3. The structure of the paper
In Sec. 2 we study the systems with detailed balance, their multiscale limits and the limit systems which
satisfy the extended principle of detailed balance. The classical Wegscheider identities for the reaction rate
constants are presented. Their limits when some of the equilibria tend to zero give the extended principle
of detailed balance.
We use the generalized mass action law for the reaction rates. For the analysis of equilibria for the general
systems, the formulas with activities are the same as for the ideal systems and it is convenient to work with
activities unless we need to study dynamics. The dynamical variables are amounts and concentrations. In
a special subsection 2.3 we discuss the relations between concentration and activities, formulate the main
assumptions and present formulas for the dissipation rate.
We introduce attractors of the systems with some irreversible reactions and study them in Sec. 3. It
includes the central results of the paper. We fully characterize the faces of the positive orthant that include
ω-limit sets. On such a face, dynamics is completely degenerated (zero rates) or it is driven by a smaller
reversible system that obeys classical thermodynamics.
Hydrogen combustion is the most studied and very important gas reaction. It serves as a main benchmark
example for many studies of chemical kinetics. This is already not a toy example but the complexity of
this system is not extremely high: in the usual models there are 6-8 components and ∼15-30 elementary
reversible reactions. Under various conditions some of these reactions are practically irreversible. We use
this system as a benchmark in Sec. 4 and give an example of the correct separation of the reactions into
reversible and irreversible part. The limit behavior of this system in time is described.
In Conclusion we briefly discuss the results with focus on the unsolved problems.
2. Multiscale limit of a system with detailed balance
2.1. Two classical approaches to the detailed balance condition
There are two traditional approach to the description of the reversible systems with detailed balance.
First, we can start from the independent rate constants of the elementary reactions and consider the solvability of the detailed balance equations as the additional condition on the admissible values of the rate
constants. Here, for m elementary reactions we have m constants (m should be an even number, m = 2ℓ,
ℓ = m/2) and some equations which describe connections between these constants. This approach was
introduced by Wegscheider in 1901 [23] and developed further by many authors [20, 4].
Secondly, we can select a “forward” reaction in each pair of mutually reverse elementary reactions. If
a positive equilibrium is known then we can find the reaction rate constants for the reverse reaction from
the constants for forward reaction and the detailed balance equations. Therefore, the forward reaction rate
constants and a set of the equilibrium activities form the complete description of the reaction. Here we
have ℓ + n independent constants, ℓ = m/2 rate constants of forward reactions and n (it is the number of
components) equilibrium activities. For these ℓ + n constants, the principle of detailed balance produces no
restrictions. This second approach is popular in applied chemical thermodynamics and kinetics [17, 10, 25]
because it is convenient to work with the independent parameters “from scratch”.
The Wegscheider conditions appear as the necessary and sufficient conditions of solvability of the detailed
balance equations. (See, for example, the textbook [24]). Let us join the forward and reverse elementary
reactions and write
X
X
βrj Aj (r = 1, . . . , ℓ) .
(5)
αri Ai ⇋
i
j
The stoichiometric matrix is Γ = (γri ), γri = βri − αri (gain minus loss). The stoichiometric vector γr is
the rth row of Γ with coordinates γri = βri − αri .
Both sides of the detailed balance equations, wr+ = wr− , are positive for positive activities. The solvability
of this system for positive activities means the solvability of the following system of linear equations:
X
γri xi = ln kr+ − ln kr− = ln Kr (r = 1, . . . ℓ)
(6)
i
4
+
−
(xi = ln aeq
i ). Of course, we assume that if kr > 0 then kr > 0 (reversibility) and the equilibrium constant
Kr > 0 is defined for all reactions from (5).
Proposition 1. The necessary and sufficient conditions for existence of the positive equilibrium aeq
i > 0
with detailed balance is: For any solution λ = (λr ) of the system
!
ℓ
X
λΓ = 0
i.e.
λr γri = 0 for all i
(7)
r=1
the Wegscheider identity holds:
ℓ
Y
(kr+ )λr =
ℓ
Y
(kr− )λr .
(8)
r=1
r=1
It is sufficient to use in (8) any basis of solutions of the system (7): λ ∈ {λ1 , · · · , λq }.
2.2. Multiscale degeneration of equilibria
We consider the systems with some irreversible reactions as the limits of the fully reversible systems
when some reaction rate constants tend to zero. In the reversible systems, the principle of detailed balance
implies the Wegscheider identities (8). Therefore, the limit system is not arbitrary. Some consequences of
the Wegscheider identities persist though a part of reaction rate constants in these identities become zero. In
[12] we compare these consequences with the grin of the Cheshire cat: the whole cat (the reversible system
with detailed balance) vanishes but the grin persists.
We can postulate that some reaction rate constants go to zero. However, the reaction rate constants
are not independent. They are connected by the Wegscheider identities. The rate constants should tend to
zero with preservation of their relations. Therefore, the simple strategy, just to neglect the rates of some
of the reactions, cannot be applied for complex reactions. Nevertheless, we can change the variables and
use the independent set “reaction rate constants for the forward reactions + equilibrium activities” (see
[17, 10, 25, 12]). Every set of positive values of these variables corresponds to a reversible system with
detailed balance and no additional restrictions are needed. If the reversible system degenerates to a system
with some irreversible reactions then some of the equilibrium activities tend to zero. Let us study this process
of degeneration of reversible reactions into irreversible ones starting from the corresponding degeneration of
equilibrium activities to zero.
Let us take a system with detailed balance and send some of the equilibrium activities to zero: aeq
i → 0
when i ∈ I for some set of indexes I. Immediately we find a surprise: this assumption is not sufficient to
find a limiting irreversible mechanism. It is necessary to take into account the rates of the convergency to
zero of different aeq
i . Indeed, let us study a very simple example,
k1
k2
k−1
k−2
A1 ⇋ A2 ⇋ A3
eq
when aeq
1 , a2 → 0.
k1
eq
eq
eq
eq
If aeq
1 , a2 → 0, a1 /a2 = const > 0 and a3 = const > 0 then the limit system should be A1 ⇋ A2 → A3
k−1
and we can keep k1,−1,2 = const whereas k−2 → 0.
eq
eq
eq
If aeq
1 , a2 → 0, a1 /a2 → 0 then the limit system should be A1 → A2 → A3 and we can keep
k1,2 = const > 0 whereas k−1,−2 → 0.
eq
eq
eq
If aeq
1 , a2 → 0, a2 /a1 → 0 then in the limit survives only one reaction A2 → A3 (if we assume that all
the reaction rate constants are bounded).
δi
We study asymptotics aeq
i = const × ε , ε → 0 for various values of non-negative exponents δi ≥ 0
(i = 1, . . . , n). At equilibrium, each reaction rate in the generalized mass action law is proportional to a
power of ε:
P
P
wreq+ = kr+ const × ε i αri δi , wreq− = kr− const × ε i βri δi .
5
According to the principle of detailed balance, wreq+ = wreq− and
kr+
= const × ε(γr ,δ) ,
kr−
(9)
where δ is the vector of exponents, δ = (δi ).
There are three groups of reactions with respect to the given vector δ:
1. (γr , δ) = 0; 2. (γr , δ) < 0; 3. (γr , δ) > 0 .
In the first group ((γr , δ) = 0) the ratio kr+ /kr− remains constant and we can take kr± = const > 0. In
the second group ((γr , δ) < 0) the ratio kr− /kr+ → 0 and we should take kr− → 0 whereas kr+ may remain
constant and positive. In the third group ((γr , δ) > 0), the situation is inverse: kr+ /kr− → 0 and we can take
kr− = const > 0, whereas kr+ → 0.
These three groups depend on δ but this dependence is piecewise constant. For every γr , three sets of δ
are defined: (i) hyperplane (γr , δ) = 0, (ii) hemispace (γr , δ) < 0 and hemispace (γr , δ) > 0. The space of
vectors δ is split in the subsets defined by the values of functions sign(γr , δ) (±1 or 0).
We consider bounded systems, hence the negative values of δ should be forbidden. At least one equilibrium activity should not vanish. Therefore, δj = 0 for some j. Below we assume that δi ≥ 0 and δj = 0
for a non-empty set of indices J0 . Moreover, the atom balance in equilibrium should be positive. Here,
this means that for the set of equilibrium concentrations ceq
i (i ∈ J0 ) the corresponding values of all atomic
concentrations are strictly positive and separated from zero.
Let the vector of exponents, δ = (δi ) be given and the three groups of reactions be found. For the
reactions of the third group (with (γr , δ) > 0) the forward reaction vanishes in the limit ε → 0. It is
convenient to transpose the stoichiometric equations for these reactions and swap the forward reactions
with reverse ones. Let us perform this transposition. After that, αr changes over βr , γ transforms into −γ,
and the inequality (γr , δ) > 0 transforms into (γr , δ) < 0.
Let us summarize. We use the given vector of exponents δ and produce a system with some irreversible
reactions from a system of reversible reactions and detailed balance equilibrium aeq
i by the following rules:
1.
2.
3.
4.
eq
if δi > 0 then we assign aeq
i = 0 and if δi = 0 then ai does not change;
±
if (γr , δ) = 0 then kr do not change;
if (γr , δ) < 0 then we assign kr− = 0 and kr+ does not change;
if (γr , δ) > 0 then we assign kr+ = 0 and kr− does not change. (In the last case, we transpose the
stoichiometric equation and swap the forward reaction with reverse one, for convenience, γr changes
to -γr and kr− becomes 0. Therefore, this case transforms into case 3.)
This is a limit system caused by the multiscale degeneration of equilibrium. The multiscale character of the
δi
→ 0 (for some i) is important because for different values of δ reactions may have
limit aeq
i = const × ε
different dominant directions and the set of irreversible reactions in the limit may change.
The general form of the kinetic equations for the homogeneous systems is
X
dN
=V
wr γr ,
dt
r
(10)
where Ni is the amount of Ai , N is the vector with components Ni and V is the volume.
Let us consider a limit system for the degeneration of equilibrium with the vector of exponents δ. For
this system (γr , δ) ≤ 0 for all r and, in particular, (γr , δ) < 0 for all irreversible reactions and (γr , δ) = 0 for
all reversible reactions.
Proposition 2. A linear functional Gδ (N ) = (δ, N ) decreases along the solutions of kinetic equations
(10) for this limit system: dGδ (N )/dt ≤ 0 and dGδ (N )dt = 0 if and only if all the reaction rates for the
irreversible reactions are zero.
6
Proof. Indeed,
X
dGδ (N )
=V
wr (γr , δ) ≤ 0 ,
dt
r
(11)
because for reversible reactions (γr , δ) = 0, and for irreversible reactions wr = wr+ ≥ 0 and (γr , δ) < 0. All
the terms in this sum are non-negative, hence it may be zero if and only if each summand is zero.
This Lyapunov function may be used in a proof that the rates of all irreversible reactions in the system
tend to 0 with time. Indeed, if they do not tend to zero then on a solution of (10) Gδ (N (t)) → −∞ when
t → ∞ and N (t) is unbounded. Equation (11) and Proposition (2) give us the possibility to prove the
extended principle of detailed balance in the following form. Let us consider a reaction mechanism that
includes reversible and irreversible reactions. Assume that the reaction rates satisfy the generalized mass
action law (3) and the set of reaction rate constants is given. Let us ask the question: Is it possible to obtain
this reaction mechanism and reaction rate constants as a limit in the multiscale degeneration of equilibrium
from a fully reversible system with the classical detailed balance. The answer to this question gives the
following theorem about the extended principle of detailed balance.
Theorem 1. A system can be obtained as a limit in the multiscale degeneration of equilibrium from a
reversible system with detailed balance if and only if (i) the reaction rate constants of the reversible part
of the reaction mechanism satisfy the classical principle of detailed balance and (ii) the convex hull of the
stoichiometric vectors of the irreversible reactions does not intersect the linear span of the stoichiometric
vectors of reversible reactions.
Proof. Let the given system be a limit of a reversible system with detailed balance in the multiscale degeneration of equilibrium with the exponent vector δ. Then for the reversible reactions (γr , δ) = 0 and for the
irreversible reactions (γr , δ) < 0. For every vector x from the convex hull of the stoichiometric vector of the
irreversible reactions (x, δ) < 0 and for any vector y from the linear span of the stoichiometric vectors of
the reversible reactions (y, δ) = 0. Therefore, these sets do not intersect. The reaction rate constants for
the reversible reactions satisfy the classical principle of detailed balance because they do not change in the
equilibrium degeneration and keep this property of the original fully reverse system with detailed balance.
Conversely, let a system satisfy the extended principle of detailed balance: (i) the reaction rate constants
of the reversible part of the reaction mechanism satisfy the classical principle of detailed balance and (ii)
the convex hull of the stoichiometric vectors of the irreversible reactions does not intersect the linear span
of the stoichiometric vectors of reversible reactions. Due to the classical theorems of the convex geometry,
there exists a linear functional that separates this convex set from the linear subspace. (Strong separation
of closed and compact convex sets.) This separating functional can be represented in the form (x, θ) for
some vector θ. For the reversible reactions (γr , θ) = 0 and for the irreversible reactions (γr , θ) < 0.
It is possible to find vector δ with this separation property and non-negative coordinates. Indeed,
according to the basic assumptions, there exists a linear conservation law with strongly positive coordinates.
This is a vector b (bi > 0) with the property: (γr , b) = 0 for all reactions. For any λ, the vector θ + λb
has the same separation property as the vector θ has. We can select such λ that δi = θi + λbi ≥ 0 and
δi = θi + λbi = 0 for some i. Let us take this linear combination δ as a vector of exponents.
Let us create a fully reversible system from the initial partially irreversible one. We do not change the
reversible reactions and their rate constants. Because the reversible reactions satisfy the classical principle
of detailed balance, there exists a strongly positive vector of equilibrium activities a∗i > 0 for the reversible
reactions.
For each irreversible reaction with the stoichiometric vector γr and reaction rate constant kr = kr+ > 0
we add a reverse reaction with the reaction rate constant
Y
kr− = kr+ (a∗i )−γri .
i
a∗i
> 0 provide the point of detailed balance. In the multiscale
For this fully reversible system the activities
∗ δi
degeneration process, the equilibrium activities depend on ε → 0 as aeq
i = ai ε . For the reactions with
7
ଵ ଶ
κ
•’ƒሼߛ
ଵ ǡ ߛଶ ǡ ǥ ߛκ ሽ
௝
଴
…‘˜ሼߛҧ௥ ȁ‫ܬ א ݎ‬ଵ ሽ
a
ߛଶ ǡ ǥ ߛκ ሽ
Թ௡
•’ƒሼߛଵ ǡ ߛଶ ǡ ǥ ߛκ ሽ
b
…‘˜ሼߛ௥ ȁ‫ܬ א ݎ‬ଵ ሽ
ሼߛ௥ ȁ‫ܬ א ݎ‬ଵ ሽ
൛ߛ௝ ห݆ ‫ܬ א‬଴ ൟ
•’ƒሼߛଵ ǡ ߛଶ ǡ ǥ ߛκ ሽ
c
b
d
•’ƒሼߛଵ ǡ ߛଶ ǡ ǥ ߛκ ሽ/•’ƒ൛ߛ௝ ห݆ ‫ܬ א‬଴ ൟ
…‘˜ሼߛҧ௥ ȁ‫ܬ א ݎ‬ଵ ሽ
ሼߛ௥ ȁ‫ܬ א ݎ‬ଵ ሽ
•’ƒ൛ߛ௝ ห݆ ‫ܬ א‬଴ ൟ
•’ƒሼߛଵ ǡ ߛଶ ǡ ǥ ߛκ ሽ/•’ƒ൛ߛ௝ ห݆ ‫ܬ א‬଴ ൟ
…‘˜ሼߛҧ௥ ȁ‫ܬ א ݎ‬ଵ ሽ
•’ƒሼߛଵ ǡ ߛଶ ǡ ǥ ߛκ ሽ
…‘˜ሼߛ௥ ȁ‫ܬ א ݎ‬ଵ ሽ
Figure 1: Main operations in the application of the extended detailed balance conditions. In the concentration space Rn we
should find the subspace spanned by all the stoichiometric vectors {γr | r = 1, . . . , ℓ} (a). In this subspace we have to select
the internal coordinates. In span{γr | r = 1, . . . , ℓ} we have to select the subspace spanned by the stoichiometric vectors of the
൛ߛ௝ ห݆ ‫ܬ א‬଴ ൟ
reversible reactions
(b) (the dashed vectors). The stoichiometric vectors of the irreversible reactions are in (b,c,d) solid and bold.
Due to the extended principle of •’ƒ൛ߛ
detailed
balance, span{γr | r ∈ J0 } should not intersect conv{γr | r ∈ J1 } (the dotted triangle in
௝ ห݆ ‫ܬ א‬଴ ൟ
Fig.). For analysis of this intersection, it is convenient to proceed to the quotation space span{γr | r = 1, . . . , ℓ}/span{γr | r ∈ J0 }
(c,d). In this quotation space, span{γ̄r | r ∈ J0 } is {0} and two situations are possible: (c) 0 ∈ conv{γ̄r | r ∈ J1 } (the dotted
triangle includes the origin) or (d) 0 ∈
/ conv{γr | r ∈ J1 } (the dotted triangle does not include the origin). In the case (c) the
extended detailed balance condition is violated. The case (d) satisfies this condition.
(γr , δ) = 0 the reaction rate constants do not depend on ε and for the reactions with (γr , δ) < 0 the rate
constant kr− tends to zero as ε−(γr ,δ) and kr+ does not change. We return to the initial system of reactions
in the limit ε → 0.
This is a particular form of the extended principle of detailed balance. For more discussion see [12]. Fig. 1
d geometric sense of the extended detailed balance condition: the convex hull of the stoichiometric
illustrates
•’ƒሼߛ ǡ ߛଶ ǡ ǥ ߛκ ሽ •’ƒ൛ߛ௝ ห݆ ‫ܬ א‬଴ ൟ
vectors of the irreversible ଵreactions
does not intersect the linear span of the stoichiometric vectors of the
reversible reactions. In this illustration, {γr | r ∈ J0 } are the stoichiometric vectors of the reversible reactions
and {γr | r ∈ J1 } are the
stoichiometric vectors of the irreversible reactions.
…‘˜ሼߛҧ ȁ‫ ܬ א ݎ‬ሽ
௥
ଵ
2.3. Activities, concentrations and affinities
To combine the linear Lyapunov functions Gδ (N ) = (δ, N ) (11) with the classical thermodynamic potential and study the kinetic equations in the closed form we have to specify the relations between activities
and concentrations. We accept the assumption: ai = ci gi (c, T ), where gi (c, T ) > 0 is the activity coefficient.
It is a continuously differentiable function of c, T in the whole diapason of their values. In a bounded region
of concentrations and temperature we can always assume that gi > g0 > 0 for some constant g0 . This
assumption is valid for the non-ideal gases and for liquid solutions. It holds also for the “surface gas” in
kinetics of heterogeneous catalysis [24] and does not hold for the solid reagents (see for example, analysis of
carbon activity in the methane reforming [12]).
The system of units should be commented. Traditionally, ai is assumed to be dimensionless and for
perfect systems ai = ci /c◦i , where c◦i is an arbitrary “standard” concentration. To avoid introduction of
unnecessary quantities, we always assume that in the selected system of units, c◦i ≡ 1.
8
If the thermodynamic potentials exist then due to the thermodynamic definition of activity (4), this
hypothesis is equivalent to the logarithmic singularity of the chemical potentials, µi = RT ln ci + . . . where
. . . stands for a continuous function of c, T (all the concentrations and the temperature). In this case, the
free energy has the form
X
Ni (ln ci − 1 + f0i (c, T )) ,
(12)
F (N, T, V ) = RT
i
where the functions f0i (c, T ) are continuously differentiable for all possible values of arguments. Functions
f0i in the right hand side of the representation (12) cannot be restored unambiguously from the free energy
function F (N, T, V ) but for a small admixture Ai it is possible to introduce the partial pressure pi which
satisfies the law pi = RT ci + o(ci ). This is due to the terms Ni ln ci in F . Indeed, P = −∂F (N, T, V )/∂V =
RT ci + o(ci ) + P |ci =0 . Connections between the equation of state, free energy and kinetics are discussed in
more detail in [7, 8].
There are several simple algebraic corollaries P
of the assumed
P connection between activities and concentrations. Let us consider an elementary reaction
αi Ai →
βi Ai with αi , βi ≥ 0. Then, according to the
generalized mass action law, for any vector of concentrations c (ci ≥ 0)
1. If, for some i, ci = 0 then γi w(c) ≥ 0;
2. If, for some i, ci = 0 and γi < 0 then αi > 0 and w(c) = 0.
P
P
Similarly, for a reversible reaction
αi Ai ⇋ βi Ai
1. If, for some i, ci = 0 and γi > 0 then βi > 0 and w− (c) = 0;
2. If, for some i, ci = 0 and γi < 0 then αi > 0 and w+ (c) = 0.
These statements as well as Proposition 3 and Corollary 1 below are the consequences of the generalized
mass action law (3) and the connection between activities and concentrations without any assumptions
about extended principle of detailed balance.
Each set of indexes J = {i1 , . . . , ij } defines a face of the positive polyhedron,
FJ = {c | ci ≥ 0 for all i and ci = 0 for i ∈ J} .
By definition, the relative interior of FJ , ri(FJ ), consists of points with ci = 0 for i ∈ J and ci > 0 for i ∈
/ J.
Proposition 3. Let for a point c ∈ ri(FJ ) and an index i ∈ J
X
γri wr (c) = 0 .
r
Then this identity holds for all c ∈ FJ .
Proof. For convenience, let us write all the forward and reverse
reactions separately and represent the
P
reaction mechanism in the form (2). All the terms in the sum r γri wr (c) are non-negative, because ci = 0.
Therefore, if the sum is zero then all the terms are zero. The reaction rate wr (3) with non-zero rate constant
takes zero value if and only if αrj > 0 and aj = 0 for some j. The equality ai = 0 is equivalent to ci = 0.
Therefore, wr (c) = 0 for a point c ∈ ri(FJ ) if and only if there exists j ∈ J such that αrj > 0. If αrj > 0
for an index j ∈ J then wr (c) = 0 for all c ∈ FJ because cj = 0 in FJ .
n
P We call a face FJ of the positive orthant R+ invariant with respect to a set S of elementary reactions if
r∈S γrj wr (c) = 0 for all c ∈ FJ and every j ∈ J.
Let us consider the reaction mechanism in the form (2) where all the forward and reverse reactions
participate separately.
Corollary 1. The following statements are equivalent:
P
1.
r∈S γri wr (c) = 0 for a point c ∈ ri(FJ ) and all indexes i ∈ J;
2. The face FJ is invariant with respect to the set of reactions S;
9
3. The face FJ is invariant with respect to every elementary reaction from S;
4. For every r ∈ S either γrj = 0 for all j ∈ J or αrj > 0 for some j ∈ J.
We aim to perform the analysis of the asymptotic behavior of the kinetic equations in the multiscale
degeneration of equilibrium described in Sec. 2.2. For this purpose, we have to answer the question: how
the relations between activities ai and concentrations ci depend on the degeneration parameter ε → 0? We
do no try to find the maximally general appropriate answer to this question. For the known applications,
the answer is: the relations between ai and ci do not depend on ε → 0. In particular, it is trivially true
for the ideal systems. The simple generalization, ai = ci gi (c, T, ε), where gi (c, T, ε) > g0 > 0 are continuous
functions, is not a generalization at all, because we can use for ε → 0 the limit case that does not depend
on ε, gi (c, T ) = gi (c, T, 0).
This independence from ε implies that the reversible part of the reaction mechanism has the thermodynamic Lyapunov functions like free energy. If we just delete the irreversible part then the classical
thermodynamics is applicable and the thermodynamic potentials do not depend on ε. For the generalized
mass action law, the time derivative of the relevant thermodynamic potentials have very nice general form.
Let, under given condition, the function Φ(N, . . .) be given, where by . . . is used for the quantities that do
not change in time under these conditions. It is the thermodynamics potential if ∂Φ(N, . . .)/∂Ni = µi . For
example, it is the free Helmholtz energy F for V, T = const and the free Gibbs energy G for P, V = const.
Let us calculate the time derivative of Φ(N, . . .) due to kinetic equation (10). The reaction rates are given
by the generalized mass action law (3) with definition of activities through chemical potential (4). We assume
that the principle of detailed balance holds (it should hold for the reversible part of the reaction mechanism
according to the extended detailed balance conditions). More precisely, there exists an equilibrium with
detailed balance for any temperature T , aeq (T ): for all r, wr+ (aeq ) = wr− (aeq ) = wreq (T ). It is convenient to
represent the reaction rates using these equilibrium fluxes wreq (T ):
!
!
X βri (µi − µeq )
X αri (µi − µeq )
−
eq
+
eq
i
i
, wr = wr exp
.
wr = wr exp
RT
RT
i
i
eq
where µeq
i = µi (a , T ).
These formulas give immediately the following representation of the dissipation rate
dΦ X ∂Φ(N, . . .) dNi X dNi
µi
=
=
dt
∂Ni
dt
dt
i
i
X
= −V RT
(ln wr+ − ln wr− )(wr+ − wr− ) ≤ 0 .
(13)
r
The inequality holds because ln is a monotone function and, hence, the expressions ln wr+ − ln wr− and
wr+ − wr− have always the same sign. Formulas of this kind for dissipation are well known since the famous
Boltzmann H-theorem (1873 [2], see also [13]). The entropy increase in isolated systems has the similar
form:
X
dS
=VR
(ln wr+ − ln wr− )(wr+ − wr− ) ≥ 0 .
dt
r
Let us notice that
ln wr+ − ln wr− =
(γr , µ)
1 X
µi (αri − βri ) = −
.
RT i
RT
The quantity −(γr , µ) is one of the central notion of physical chemistry, affinity [5]. It is positive if the
forward reaction prevails over reverse one and negative in the opposite case. It measures the energetic
advantage of the forward reaction over the reverse one (free energy per mole). The activity divided by RT
shows how large is this energetic advantage comparing to the thermal energy. We call it the normalized
affinity and use a special notation for this quantity:
(γr , µ)
RT
10
Ar = −
Let us apply an elementary identity
exp a − exp b = (exp a + exp b) tanh
a−b
2
to the reaction rate, wr = wr+ − wr− :
wr = (wr+ + wr− ) tanh
Ar
.
2
(14)
This representation of the reaction rates gives immediately for the dissipation rate:
X
Ar
dΦ
= −V RT
≤ 0.
(wr+ + wr− )Ar tanh
dt
2
r
(15)
In this formula, the kinetic information is collected in the non-negative factors, the sums of reaction rates
(wr+ + wr− ). The purely thermodynamical multipliers Ar tanh(Ar /2) are also non-negative. For small |Ar |,
the expression Ar tanh(Ar /2) behaves like A2r /2 and for large |Ar | it behaves like the absolute value, |Ar |.
So, we have two Lyapunov functions for two fragments of the reaction mechanism. For the reversible
part, this is just a classical thermodynamic potential. For the irreversible part, this is a linear functional
Gδ (N ) = (δ, N ). More precisely, the irreversible reactions decrease this functional, whereas for the reversible
reactions it is the conservation law. Therefore, it decreases monotonically in time for the whole system.
3. Attractors
3.1. Dynamical systems and limit points
The kinetic equations (10) do not give a complete representation of dynamics. The right hand side
includes the volume V and the reaction rates wr which are functions (3) of the concentrations c and temperature T , whereas in the left hand side there is Ṅ . To close this system, we need to express V , c and T
through N and quantities which do not change in time. This closure depends on conditions. The simplest
expressions appear for isochoric isothermal conditions: V, T = const, c = N/V . For other classical conditions (U, V = const, or P, T = const, or H, P = const) we have to use the equations of state. There may be
more sophisticated closures which include models or external regulators of the pressure and temperature,
for example.
Proposition 2 is valid for all possible closures. It is only important that the external flux of the chemical
components is absent. Further on, we assume that the conditions are selected, the closure is done, the right
hand side of the resulting system is continuously differentiable and there exists the positive bounded solution
for initial data in Rn+ and V , T remain bounded and separated from zero. The nature of this closure is not
crucial. For some important particular closures the proofs of existence of positive and bounded solutions are
well known (see, for example, [22]). Strictly speaking, such a system is not a dynamical system in Rn+ but a
semi-dynamical one: the solutions may lose positivity and leave Rn+ for negative values of time. The theory
of the limit behavior of the semi-dynamical systems was developed for applications to kinetic systems [9].
We aim to describe the limit behavior of the systems as t → ∞. Under the extended detailed balance
condition the limit behavior is rather simple and the system will approach steady states but to prove this
behavior we need the more general notion of the ω-limit points.
By the definition, the ω-limit points of a dynamical system are the limit points of the motions when time
t → ∞. We consider a kinetic system in Rn+ . In particular, for each solution of the kinetic equations N (t)
the set of the corresponding ω-limit points is closed, connected and consists of the whole trajectories ([9],
Proposition 1.5). This means that the motion which starts from an ω-limit point remains in Rn+ for all time
moments, both positive and negative.
Proposition 4. Let N (t) be a positive solution of the kinetic equation and x∗ be an ω-limit point of this
solution and x∗i = 0. then at this point ẋi |x∗ = 0.
11
Proof. Let x(t) be a solution of the kinetic equations with the initial state x(0) = x∗ . All the points x(t)
(−∞ < t < ∞) belong to Rn+ . Indeed, there exists such a sequence tj → ∞ that N (tj ) → x∗ . For any
τ ∈ (−∞, ∞), N (tj + τ ) → x(τ ). For sufficiently large j, tj + τ > 0 and the value N (tj + τ ) ∈ Rn+ . Therefore,
x(τ ) ∈ Rn+ (−∞ < τ < ∞) and for any τ the point x(τ ) is an ω-limit point of the solution N (t). Let x∗i = 0
and ẋi |x∗ = v 6= 0. If v > 0 then for small |τ | and τ < 0 the value of xi becomes negative, xi (τ ) < 0. It
is impossible because positivity. Similarly, If v < 0 then for small τ > 0 the value of xi becomes negative,
xi (τ ) < 0. It is also impossible because positivity. Therefore, ẋi |x∗ = 0.
We use Proposition 4 in the following combination with Proposition 3. Let us write the reaction mechanism in the form (2).
Corollary 2. If an ω-limit point belongs to the relative interior riFJ of the face FJ ⊂ Rn+ then the face FJ
is invariant with respect to the reaction mechanism and for every elementary reaction either γrj = 0 for all
j ∈ J or αrj > 0 for some j ∈ J.
Proof. If an ω-limit point belongs to riFJ then at this point all ċj = 0 for j ∈ J due to Proposition 4.
Therefore, we can apply Corollary 1.
3.2. Steady states of irreversible reactions
Under extended detailed balance conditions, all the reaction rates of the irreversible reactions are zero
at every limit point of the kinetic equations (10), due to Proposition 2. In this section, we give a simple
combinatorial description of steady states for the set of irreversible reactions. This description is based on
Proposition 2 and, therefore, uses the extended detailed balance conditions.
We continue to study multiscale degeneration of a detailed balance equilibrium. The vector of exponents
δ = (δi ) is given, δi ≥ 0 for all i and δi = 0 for some i. There are two sets of reaction. For one of them,
(γr , δ) = 0 and in the limit both kr± > 0. In the second set, (γr , δ) < 0 and in the limit we assign kr− = 0 and
kr+ is the same as in the initial system (before the equilibrium degeneration). If it is necessary, we transpose
the stoichiometric equations and swap the forward reactions with reverse ones.
For convenience, let us change the notations. Let γi be the stoichiometric vectors of reversible reactions
with (γr , δ) = 0 (r = 1, . . . , h), and νl be the stoichiometric vectors for the reactions from the second set,
(νl , δ) < 0 (l = 1, . . . , s). For the reaction rates and constants for the first set we keep the same notations:
wr , wr± , kr± . For the second set, we use for the reaction rate constants ql = ql+ and for the reaction rates
vl = vl+ . (They are also calculated according to the generalized mass action law (3).) The input and output
stoichiometric coefficients remain αri and βri for the first set and for the second set we use the notations
ανli and βliν .
Let the rates of all the irreversible reaction be equal to zero. This does not mean that all the concentrations ai with δi > 0 achieve zero. A bimolecular reaction A + B → C gives us a simple example:
w = kaA aB and w = 0 if either aA = 0 or aB = 0. On the plane with coordinates aA , aB and with the
positivity condition, aA , aB ≥ 0, the set of zeros of w is a union of two semi-axes, {aA = 0, aB ≥ 0} and
{aA ≥ 0, aB = 0}. In more general situation, the set in the activity space, where all the irreversible reactions
have zero rates, has a similar structure: it is the union of some faces of the positive orthant.
P
Let us describe the set of the steady states of the irreversible reactions. Due to Proposition 2, if l vl νl =
0 then all vl = 0. Let us describe the set of zeros of all vl in the the positive orthant of activities.
For every l = 1, . . . , s the set of zeros of vl in Rn+ is given by the conditions: at least for one i ανli 6= 0
and ai = 0. It is convenient to represent this condition as a disjunction. Let W
Jl = {i | ανli 6= 0}. Then the set
of zeros of vl an a positive orthant of activities is presented by the formula i∈Jl (ai = 0). The set of zeros
of all vl is represented by the following conjunction form
∧sl=1 (∨i∈Jl (ai = 0)) .
(16)
To transform it into the unions of subspaces we have to move to a disjunction form and make some cancelations. First of all, we represent this formula as a disjunction of conjunctions:
∧sl=1 (∨i∈Jl (ai = 0)) = ∨i1 ∈J1 ,...,is ∈Js ((ai1 = 0) ∧ . . . ∧ (ais = 0)) .
12
(17)
For a cortege of indexes {i1 , . . . , is } the correspondent set of their values may be smaller because some values
il may coincide. Let this set of values be S{i1 ,...,is } . We can delete from (17) a conjunction (ai1 = 0) ∧ . . . ∧
(ais = 0) if there exists a cortege {i′1 , . . . , i′s } (i′l ∈ Jl ) with smaller set of values, S{i1 ,...,is } ⊇ S{i′1 ,...,i′s } . Let
us check the corteges in some order and delete a conjunction from (17) if there remain a term with smaller
(or the same) set of index values in the formula. We can also substitute in (17) the corteges by their sets
of values. The resulting minimized formula may become shorter. Each elementary conjunction represents a
coordinate subspace and after cancelations each this subspace does not belong to a union of other subspaces.
The final form of formula (17) is
(18)
∨j (∧i∈Sj (ai = 0)) ,
where Sj are sets of indexes, Sj ⊂ {1, . . . , n} and for every two different Sj , Sp none of them includes
another, Sj * Sp . The elementary conjunction ∧i∈Sj (ai = 0) describes a subspace.
The steady states of the irreversible part of the reaction mechanism are given by the intersection of
the union of the coordinate subspaces (18) with Rn+ . For applications of this formula, it is important that
the equalities ai = 0, ci = 0 and Ni = 0 are equivalent and the positive orthants of the activities ai ,
concentrations ci or amounts Ni represent the same sets of physical states. This is also true for the faces of
these orthants: FJ for the activities, concentrations or amounts correspond to the same sets of states. (The
same state may corresponds to the different points of these cones, but the totalities of the states are the
same.)
3.3. Sets of steady states of irreversible reactions invariant with respect to reversible reactions
In this Sec. we study the possible limit behavior of systems which satisfy the extended detailed balance
conditions and include some irreversible reactions. All the ω-limit points of such systems are steady states
of the irreversible reactions due to Proposition 2 but not all these steady states may be the ω-limit points
of the system. A simple formal example gives us the couple of reaction: A ⇋ B, A + B → C. Here,
we have one reversible and one irreversible reaction. The conditions of the extended detailed balance hold
(trivially): the linear span of the stoichiometric vector of the reversible reaction, (−1, 1, 0), does not include
the stoichiometric vector of the irreversible reaction, (−1, −1, 1). For the description of the multiscale
degeneration of equilibrium, we can take the exponents δA = 1, δB = 1, δC = 0.
The steady states of the irreversible reaction are given in Rn+ by the disjunction, (cA = 0) ∨ (cB = 0)
but only the points (cA = cB = 0) may be the limit points when t → ∞. Indeed, if cA = 0 and cB > 0 then
dcA /dt = k1− cB > 0. Due to Proposition 4 this is not an ω-limit point. Similarly, the points with cA > 0
and cB = 0 are not the ω-limit points.
Let us combine Propositions 2, 4 and Corollary 2 in the following statement.
Theorem 2. Let the kinetic system satisfy the extended detailed balance conditions and include some irreversible reactions. Then an ω-limit point x∗ ∈ riFJ exists if and only if FJ consists of steady states of the
irreversible reactions and is invariant with respect to all reversible reactions.
Proof. If an ω-limit point x∗ ∈ riFJ exists then it is a steady state for all irreversible reactions (due to
Propositions 2). Therefore, the face FJ consists of steady-states of the irreversible reactions (Proposition 4)
and is invariant with respect to all reversible reactions (Proposition 4 and Corollary 2). To prove the reverse
statement, let us assume that FJ consists of steady states of the irreversible reactions and is invariant with
respect to all reversible reactions. The reversible reactions which do not act on cj for j ∈ J define a semidynamical system on FJ . The positive conservation law b defines an positively invariant polyhedron in FJ .
Dynamics in such a compact set always has ω-limit points.
Let us find the faces FJ that contain the ω-limit points in their relative interior riFJ . According
to Theorem 2, these faces should consist of the steady states of the irreversible reactions and should be
invariant with respect to all reversible reactions. Let us look for the maximal faces with this property. For
this purpose, we always minimize the disjunctive forms by cancelations. We do not list the faces that contain
the ω-limit points in their relative interior and are the proper subsets of other faces with this property. All
the ω-limit points belong to the union of these maximal faces.
13
Let us start from the minimized disjunctive form (18). Equation (18) represents the set of the steady
states of the irreversible part of the reaction mechanism by a union of the coordinate subspaces ∧i∈Sj (ci = 0)
in intersection with Rn+ . It is the union of the faces, ∪j FSj . If a face FJ consists of the steady states of the
irreversible reactions then J ⊇ Sj for some j.
The following formula (19) is true on a face FJ if it contains ω-limit points in the relative interior riFJ
(Theorem 2):
(ci = 0) ⇒ ∧r,γri >0 ∨j,αrj >0 (cj = 0) ∧ ∧r,γri <0 ∨j,βrj >0 (cj = 0) .
(19)
Here, ci = 0 in FJ may be read as i ∈ J. Following the previous section, we use here the notations γri , βri
and βri for the reversible reactions and reserve νl , ανli and βliν for the irreversible reactions. The set of γr in
this formula is the set of the stoichiometric vectors of the reversible reactions.
The required faces FJ may be constructed in an iterative procedure. First of all, let us introduce an
operation that transforms a set of indexes S ⊂ {1, 2, . . . , n} in a family of sets, S(S) = {S1′ , . . . , Sl′ }. Let
us take formula (19) and find the set where it is valid for all i ∈ S. This set is described by the following
formula:
∧i∈S (ci = 0) ∧ ∧r,γri >0 ∨j,αrj >0 (cj = 0) ∧ ∧r,γri <0 ∨j,βrj >0 (cj = 0) .
(20)
Let us produce a disjunctive form of this formula and minimize it by cancelations as it is described in
Sec. 3.2. The result is
(21)
∨j=1,...,k ∧i∈Sj′ (ci = 0) .
Because of cancelations, the sets Sj′ do not include one another. They give the result, S(S) = {S1′ , . . . , Sl′ }.
Each Sj′ ∈ S(S) is a superset of S, S ′ ⊇ S.
Let us extend the operation S on the sets of sets S = {S1 , . . . , Sp } with the property: Si 6⊂ Sj for i 6= j.
Let us apply S to all Si and take the union of the results: S0 (S) = ∪i S(Si ). Some sets from this family
may include other sets from it. Let us organize cancelations: if S ′ , S ′′ ∈ S0 (S) and S ′ ⊂ S ′′ then retain the
smallest set, S ′ , and delete the largest one. We do the cancelations until it is possible. Let us call the final
result S(S). It does not depend on the order of these operations.
Let us start from any family S and iterate the operation S. Then, after finite number of iterations, the
sequence Sd (S) stabilizes: Sd (S) = Sd+1 (S) = . . . because for any set S all sets from S(S) include S.
The problems of propositional logic that arise in this and the previous section seem very similar to
elementary logical puzzles [3]. In the solution we just use the logical distribution laws (distribution of
conjunction over disjunction and distribution of disjunction over conjunction), commutativity of disjunction
and conjunction, and elementary cancelation rules like (A ∧ A) ⇔ A, (A ∨ A) ⇔ A, [A ∧ (A ∨ B)] ⇔ A, and
[A ∨ (A ∧ B)] ⇔ A.
Now, we are in position to describe the construction of all FJ that have the ω-limit points on their
relative interior and are the maximal faces with this property.
1. Let us follow Sec. 3.2 and construct the minimized disjunctive form (18) for the description of the
steady states of the irreversible reactions.
2. Let us calculate the families of sets Sd ({Sj }) for the family of sets {Sj } from (18) and d = 1, 2, . . .,
until stabilization.
3. Let Sd ({Sj }) = Sd+1 ({Sj }) = {J1 , J2 , . . . Jp }. Then the family of the faces FJi (i = 1, 2, . . . , p) gives
the answer: the ω-limit points are situated in riFJi and for each i there are ω-limit points in riFJi .
3.4. Simple examples
In this Sec., we present two simple and formal examples of the calculations described in the previous
sections.
1. A1 + A2 ⇋ A3 + A4 , γ = (−1, −1, 1, 1, 0); A1 + A2 → A5 , ν = (−1, −1, 0, 0, 1). The extended
principle of detailed balance holds: the convex hull of the stoichiometric vectors of the irreversible reactions
consists of one vector γ2 and it is linearly independent of γ1 . The input vector α for the irreversible reaction
A1 + A2 → A5 is (−1, −1, 0, 0, 0). The set J = Jl from the conjunction form (16) is defined by the non-zero
14
coordinates of this αν : J = {1, 2}. The conjunction form in this simple case (one irreversible reaction)
loses its first conjunction operation and is just (c1 = 0) ∨ (c2 = 0). It is, at the same time, the minimized
disjunction form (18) and does not require additional transformations. This formula describes the steady
states of the irreversible reaction in the positive orthant Rn+ . For this disjunction form, The family of sets
S = {Sj } consists of two sets, S1 = {1} and S2 = {2}.
Let us calculate S(S1,2 ). For both cases, i = 1, 2 there are no reversible reactions with γri = 0. Therefore,
one expression in round parentheses vanishes in (20). For S = {1} this formula gives
(c1 = 0) ∧ ((c3 = 0) ∨ (c4 = 0))
and for S = {2} it gives
(c2 = 0) ∧ ((c3 = 0) ∨ (c4 = 0)) .
The elementary transformations give the disjunctive forms:
[(c1 = 0) ∧ ((c3 = 0) ∨ (c4 = 0))] ⇔ [((c1 = 0) ∧ (c3 = 0)) ∨ ((c1 = 0) ∧ (c4 = 0))] ,
[(c2 = 0) ∧ ((c3 = 0) ∨ (c4 = 0))] ⇔ [((c2 = 0) ∧ (c3 = 0)) ∨ ((c2 = 0) ∧ (c4 = 0))] .
Therefore, S(S1 ) = {{1, 3}, {1, 4}}, S(S2 ) = {{2, 3}, {2, 4}} and
S({S1 , S2 }) = {{1, 3}, {1, 4}, {2, 3}, {2, 4}} .
No cancelations are needed. The iterations of S do not produce new sets from {{1, 3}, {1, 4}, {2, 3}, {2, 4}}.
Indeed, if c1 = c3 = 0, or c1 = c4 = 0, or c2 = c3 = 0, or c2 = c4 = 0 then all the reaction rates are zero.
More formally, for example for S({1, 3}) formula (20) gives
[(c1 = 0) ∧ ((c3 = 0) ∨ (c4 = 0))] ∧ [(c3 = 0) ∧ ((c1 = 0) ∨ (c2 = 0))] .
This formula is equivalent to (c1 = 0) ∧ (c3 = 0). Therefore, S({1, 3}) = {1, 3}. The same result is true for
{1, 4}, {2, 3}, and {2, 4}.
All the ω-limit points (steady states) belong to the faces F{1,3} = {c |, c1 = c3 = 0}, F{1,4} = {c |, c1 =
c4 = 0}, F{2,3} = {c |, c2 = c3 = 0}, or F{2,4} = {c |, c2 = c4 = 0}. The position of the ω-limit point for a
solution N (t) depends on the initial state. More specifically, this system of reactions has three independent
linear conservation laws: b1 = N1 + N2 + N3 + N4 + 2N5 , b2 = N1 − N2 and b3 = N3 − N4 . For given values
of these b1,2,3 vector N belongs to the 2D plane in R5 . The intersection of this plane with the selected faces
depends on the signs of b2,3 :
• If b2 < 0, b3 < 0 then it intersects F{1,3} only, at one point N = (0, −b2 , 0, −b3 , b1 + b2 + b3 ) (N5 should
be non-negative, b1 + b2 + b3 ≥ 0) .
• If b2 = 0, b3 < 0 then it intersects both F{1,3} and F{2,3} at one point N = (0, 0, 0, −b3, b1 + b3 ) (N5
should be non-negative, b1 + b3 ≥ 0).
• If b2 < 0, b3 = 0 then it intersects both F{1,3} and F{1,4} at one point N = (0, −b2 , 0, 0, b1 + b2 ) (N5
should be non-negative, b1 + b2 ≥ 0).
• If b2 > 0, b3 < 0 then it intersects F{2,3} only, at one point N = (b2 , 0, 0, −b3, b1 + b2 + b3 ) (N5 should
be non-negative, b1 + b2 + b3 ≥ 0).
• If b2 > 0, b3 = 0 then it intersects F{2,3} and F{2,4} at the point N = (b2 , 0, 0, 0, b1 + b2 ) (N5 is
non-negative because b1 + b2 + b3 ≥ 0).
• If b2 < 0, b3 > 0 then it intersects F{1,4} only, at one point N = (0, −b2 , b3 , 0, b1 + b2 + b3 ) (N5 should
be non-negative, b1 + b2 + b3 ≥ 0).
• If b2 = 0, b3 > 0 then it intersects F{1,4} and F{2,4} at one point N = (0, 0, b3 , 0, b1 + b3 ) (N5 is
non-negative because b1 + b3 ≥ 0).
15
Table 1: H2 burning mechanism [21]
No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Reaction
H2 + O2 ⇋ 2OH
H2 + OH ⇋ H2 O + H
OH + O ⇋ O2 + H
H2 + O ⇋ OH + H
O2 + H +M ⇋ HO2 +M
OH + HO2 ⇋ O2 + H2 O
H + HO2 ⇋ 2OH
O + HO2 ⇋ O2 + OH
2OH ⇋ H2 O + O
2H + M ⇋ H2 + M
2H + H2 ⇋ H2 + H2
2H + H2 O ⇋ H2 + H2 O
OH + H + M ⇋ H2 O + M
H + O + M ⇋ OH + M
2O + M ⇋ O2 + M
H + HO2 ⇋ H2 + O2
2HO2 ⇋ O2 + H2 O2
H2 O2 + M ⇋ 2OH + M
H + H2 O2 ⇋ H2 + HO2
OH + H2 O2 ⇋ H2 O + HO2
Stoichiometric vector
(-1,-1,2,0,0,0,0,0)
(-1,0,-1,1,1,0,0,0)
(0,1,-1,0,1,-1,0,0)
(-1,0,1,0,1,-1,0,0)
(0,-1,0,0,-1,0,1,0)
(0,1,-1,1,0,0,-1,0)
(0,0,2,0,-1,0,-1,0)
(0,1,1,0,0,-1,-1,0)
(0,0,-2,1,0,1,0,0)
(1,0,0,0,-2,0,0,0)
(1,0,0,0,-2,0,0,0)
(1,0,0,0,-2,0,0,0)
(0,0,-1,1,-1,0,0,0)
(0,0,1,0,-1,-1,0,0)
(0,1,0,0,0,-2,0,0)
(1,1,0,0,-1,0,-1,0)
(0,1,0,0,0,0,-2,1)
(0,0,2,0,0,0,0,-1)
(1,0,0,0,-1,0,1,-1)
(0,0,-1,1,0,0,1,-1)
• If b2 > 0, b3 > 0 then it intersects F{2,4} only, at one point N = (b2 , 0, b3 , 0, b1 + b2 + b3 ) (N5 is
non-negative because b1 + b2 + b3 ≥ 0).
As we can see, the system has exactly one ω-limit point for any admissible combination of the values of the
conservation laws. These points are the listed points of intersection.
For the second simple example, let us change the direction of the irreversible reaction.
2. A1 + A2 ⇋ A3 + A4 , γ1 = (−1, −1, 1, 1, 0), A5 → A1 + A2 , ν = (1, 1, 0, 0, −1). The extended principle
of detailed balance holds. The steady-states of the irreversible reactions is given by one equation, c5 = 0.
Formula (20) gives for S({5}) just (c5 = 0). The face F{5} includes ω-limit points in riF{5} . Dynamics
on this face is defined by the fully reversible reaction system and tends to the equilibrium of the reaction
A1 + A2 ⇋ A3 + A4 under the given conservation laws. On this face, there exist the border equilibria, where
c1 = c3 = 0, or c1 = c4 = 0, or c2 = c3 = 0, or c2 = c4 = 0 but they are not attracting the positive solutions.
4. Example: H2 +O2 system
For the case study, we selected the H2 +O2 system. This is one of the main model systems of gas kinetics.
The hydrogen burning gives us an example of the medium complexity with 8 components (A1 =H2 , A2 =O2 ,
A3 =OH, A4 =H2 O, A5 =H, A6 =O, A7 =HO2 , and A8 =H2 O2 ) and 2 atomic balances (H and O). For the
example, we selected the reaction mechanism from [21]. The literature about hydrogen burning mechanisms
is huge. For recent discussion we refer to [16, 18]. We do not aim to compare the different schemes of this
reaction but use this reaction mechanism as an example and a benchmark.
A special symbol “M” is used for the “third body”. It may be any molecule. The third body provides the
energy balance. Efficiency of different molecules in this process is different, therefore, the “concentration” of
the third body is a weighted sum of the concentrations of the components with positive weights. The third
body does not affect the equilibrium constants and does not change the zeros of the forward and reverse
reaction rates but modifies the non-zero values of reaction rates. Therefore, for our analysis we can omit
these terms. The elementary reactions 10, 11 and 12 are glued in one, 2H⇋H2 , after cancelation of the third
bodies, and we analyze the mechanism of 18 reaction.
16
Under various conditions, some of the reactions are (almost) irreversible and some of them should be
considered as reversible. For example, let us consider the H2 +O2 system at or near the atmospheric pressure
and in the temperature interval 800–1200K. The reactions 1, 2, 4, 18, 19, and 20 are supposed to be reversible
(on the base of the reaction rate constants presented in [21]). The first question is: if these reactions are
reversible then which reactions may be irreversible?
Due to the general criterion, the convex hull of the stoichiometric vectors of the irreversible reactions has
empty intersection with the linear span of the stoichiometric vectors of the reversible reactions. Therefore,
if the stoichiometric vector of a reaction belongs to the linear span of the stoichiometric vectors of the
reversible reactions, then this reaction is reversible. Simple linear algebra gives that
γ3,5,9 ∈ span{γ1 , γ2 , γ4 , γ18 , γ19 , γ20 } .
In particular, γ3 = −γ1 + γ4 , γ5 = γ1 − γ18 + γ19 , γ9 = γ2 − γ4 . So, the list of the reversible reactions
should include the reactions 1, 2, 3, 4, 5, 9, 18, 19, and 20. The reactions 6, 7, 8, 10, 11, 12, 13, 14, 15,
and 17 may be irreversible. Formally, there are 28 = 256 possible combinations of the directions of these
8 reactions (the reactions 10, 11 and 12 have the same stoichiometric vector and, in this sense, should be
considered as one reaction). The general criterion and simple linear algebra give that there are only two
admissible combinations of the directions of irreversible reactions: either for all of them kr− = 0 or for all of
them kr+ = 0. Here, the forward and reverse reactions and the notations kr± are selected according to the
Table 1. We can immediately notice that the inverse direction of all reactions is very far from the reality
under the given conditions, for example, it includes the irreversible dissociation H2 → 2H.
Let us demonstrate in detail, how the general criterion produces this reduction from the 256 possible
combinations of directions of irreversible reactions to just 2 admissible combinations. We assume that the
initial set of reactions is spit in two: reversible reactions with numbers r ∈ J0 and irreversible reactions with
r ∈ J1 , rank{γ1 , γ2 , . . . , γℓ } = d, rank{γr | r ∈ J0 } = d0 . The rank of all vectors γr , d, must exceed the rank
of the stoichiometric vectors of the reversible reactions, d > d0 , because if d = d0 then all the reactions must
be reversible and the problem becomes trivial.
According to [12], we have to perform the following operations with the set of stoichiometric vectors γr
(Fig. 1):
1. Eliminate several coordinates from all γr using linear conservation laws. This is transfer to the internal
coordinates in span{γr | r = 1, . . . , ℓ};
2. Eliminate coordinates from all γr (r ∈ J1 ) using the stoichiometric vectors of the reversible reactions
and the Gauss–Jordan elimination procedure. This is the map to the quotient space span{γj | j =
1, . . . , ℓ}/span{γj | j ∈ J0 }. Me denote the result as γ r ;
3. Use the linear programming technique and analyze for which combinations of the signs, the convex
hull conv{±γ r | r ∈ J1 } does not include 0.
In the Table 2 we present the results of the step-by-step elimination. First, the atomic balances give us
for every possible stoichiometric vector η = (η1 , . . . , η8 ) two identities:
1. 2η1 + η3 + 2η4 + η5 + η7 + 2η8 = 0 or η1 = − 21 (η3 + 2η4 + η5 + η7 + 2η8 );
2. 2η2 + η3 + η4 + η6 + 2η7 + 2η8 = 0 or η2 = − 21 (η3 + η4 + η6 + 2η7 + 2η8 ).
Let us recall that the order of the coordinates (η1 , . . . , η8 ) corresponds to the following order of the components, (H2 , O2 , OH, H2 O, H, O, HO2 , H2 O2 ). Due to these identities, a stoichiometric vector η for this
mixture is completely defined by six coordinates (η3 , . . . , η8 ). In the second column of the Table 2 these 6D
vectors are given for all the reactions from the H2 burning mechanism (the Table 1).
In five columns No. 3-7, the results of the coordinate eliminations are presented (and the zero-valued
eliminated coordinates are omitted). Each elimination step may be represented as a projection:
x 7→ x − xi
1
η,
ηi
where ηi is a pivot (highlighted in bold in the column preceding the result of elimination), and η is the vector
that includes the pivot (as the ith coordinate). The projection operator is applied to every vector of the
17
Table 2: Elimination of coordinates of stoichiometric vectors for H2 burning mechanism. The reversible reactions are collected
in the upper part of the Table. The reaction in the lower part of the table are irreversible. The group of equivalent reactions
10, 11, 12 is presented by one of them. In the second column, the first two coordinates (which correspond to H2 and O2 ) are
excluded using the atomic balance. In the following columns the results of the coordinates elimination are presented. For each
step, the pivot for elimination is underlined and highlighted in bold in the previous column. The eliminated coordinates at
each step are named at the top of each column. Their zero values are omitted.
No
1
2
3
4
5
9
18
19
20
6
7
8
10
13
14
15
16
17
H2 , O2
(2,0,0,0,0,0)
(-1,1,1,0,0,0)
(-1,0,1,-1,0,0)
(1,0,1,-1,0,0)
(0,0,-1,0,1,0)
(-2,1,0,1,0,0)
(2,0,0,0,0,-1)
(0,0,-1,0,1,-1)
(-1,1,0,0,1,-1)
(-1,1,0,0,-1,0)
(2,0,-1,0,-1,0)
(1,0,0,-1,-1,0)
(0,0,-2,0,0,0)
(-1,1,-1,0,0,0)
(1,0,-1,-1,0,0)
(0,0,0,-2,0,0)
(0,0,-1,0,-1,0)
(0,0,0,0,-2,1)
OH
(0,0,0,0,0)
(1,1,0,0,0)
(0,1,-1,0,0)
(0,1,-1,0,0)
(0,-1,0,1,0)
(1,0,1,0,0)
(0,0,0,0,-1)
(0,-1,0,1,-1)
(1,0,0,1,-1)
(1,0,0,-1,0)
(0,-1,0,-1,0)
(0,0,-1,-1,0)
(0,-2,0,0,0)
(1,-1,0,0,0)
(0,-1,-1,0,0)
(0,0,-2,0,0)
(0,-1,0,-1,0)
(0,0,0,-2,1)
H2 O2
(0,0,0,0)
(1,1,0,0)
(0,1,-1,0)
(0,1,-1,0)
(0,-1,0,1)
(1,0,1,0)
(0,0,0,0)
(0,-1,0,1)
(1,0,0,1)
(1,0,0,-1)
(0,-1,0,-1)
(0,0,-1,-1)
(0,-2,0,0)
(1,-1,0,0)
(0,-1,-1,0)
(0,0,-2,0)
(0,-1,0,-1)
(0,0,0,-2)
H2 O
(0,0,0)
(0,0,0)
(1,-1,0)
(1,-1,0)
(-1,0,1)
(-1,1,0)
(0,0,0)
(-1,0,1)
(-1,0,1)
(-1,0,-1)
(-1,0,-1)
(0,-1,-1)
(-2,0,0)
(-2,0,0)
(-1,-1,0)
(0,-2,0)
(-1,0,-1)
(0,0,-2)
H
(0,0)
(0,0)
(0,0)
(0,0)
(-1,1)
(0,0)
(0,0)
(-1,1)
(-1,1)
(-1,-1)
(-1,-1)
(-1,-1)
(-2,0)
(-2,0)
(-2,0)
(-2,0)
(-1,-1)
(0,-2)
O
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(0)
(-2)
(-2)
(-2)
(-2)
(-2)
(-2)
(-2)
(-2)
(-2)
previous column. At the end (the last column), all the stoichiometric vectors of the reversible reaction are
transformed into zero, and the stoichiometric vectors of the irreversible reactions with the given direction
(from the left to the right) are transformed into the same vector (−2). If we restore all the zeros, then the
corresponding 6D vector is (0, 0, 0, 0, −2, 0). We have to use the atomic balances to return to the 8D vectors.
The coordinate x7 corresponds to HO2 , x1 corresponds to H2 , and x2 corresponds to O2 , hence, 2x1 − 2 = 0
and 2x2 − 4 = 0. The restored 8D vector is (1, 2, 0, 0, 0, 0, −2, 0).
A convex combination of several copies of one vector cannot give zero. Therefore, the structural condition
of the extended principle of detailed balance holds. It holds also for the inverse direction of all the irreversible
reactions. All other distributions of directions can produce zero in the convex hull and are inadmissible. So,
we have the following list of irreversible reactions that satisfies the extended principle of detailed balance
for given reversible reactions. (We will not discuss the second list of reverse irreversible reactions because it
has not much sense for given conditions.)
6
OH + HO2 → O2 + H2 O
7
H + HO2 → 2OH
8
O + HO2 → O2 + OH
10 2H → H2
13 OH + H → H2 O
14 H + O → OH
15 2O → O2
16 H + HO2 → H2 + O2
17 2HO2 → O2 + H2 O2 .
We assume that all the reaction rate constants for the selected directions are strictly positive. The rate
of all these reaction vanish if and only if concentration of H, O and HO2 are equal to zero, c5,6,7 = 0. Indeed,
c5 = 0 if and only if w10 = 0, c6 = 0 if and only if w15 = 0, a7 = 0 if and only if w17 = 0. On the other
hand, all other reaction rates from this list are zeros if c5,6,7 = 0.
18
Let us reproduce this reasoning using formulas from Sec. 3.2. For the lth irreversible reaction, Jl is the
set of indexes i for which αli 6= 0. Let us keep for the irreversible reactions their numbers (6, 7, 8, 10, 13, 14,
15, 16, 17). For them, J6 = {3, 7}, J7 = {5, 7}, J8 = {6, 7}, J10 = {5}, J13 = {3, 5}, J14 = {5, 6}, J15 = {6},
J16 = {5, 7}, J17 = {7}.
Formula (18) gives for the steady states of the irreversible reactions:
((c3 = 0) ∨ (c7 = 0)) ∧ ((c5 = 0) ∨ (c7 = 0)) ∧ ((c6 = 0) ∨ (c7 = 0)) ∧ (c5 = 0)
∧((c3 = 0) ∨ (c5 = 0)) ∧ ((c5 = 0) ∨ (c6 = 0)) ∧ (c6 = 0) ∧ ((c5 = 0) ∨ (c7 = 0)) ∧ (c7 = 0).
It is equivalent to
(c5 = 0) ∧ (c6 = 0) ∧ (c7 = 0) .
Of course, the result is the same, the face F{5,6,7} (c5,6,7 = 0, ci ≥ 0) is the set of the steady states of all
irreversible reaction.
Let us look now on the list of reversible reactions:
1
H2 + O2 ⇋ 2OH
2
H2 + OH ⇋ H2 O + H
3
OH + O ⇋ O2 + H
4
H2 + O ⇋ OH + H
5
O2 + H ⇋ HO2
9
2OH ⇋ H2 O + O
18 H2 O2 ⇋ 2OH
19 H + H2 O2 ⇋ H2 + HO2
20 OH + H2 O2 ⇋ H2 O + HO2
If the concentration OH (c3 ) is positive then the component O is produced in the reaction 9. If the
concentrations of H2 (c1 ) and OH (c3 ) both are positive then the component H is produced in reaction 2.
If the concentrations of H2 O2 (c8 ) and OH (c3 ) both are positive then the component HO2 is produced
in reaction 2. Due to the reversible reaction 18 any of two components H2 O2 and OH produces the other
component. Moreover, the first reaction produces OH from H2 + O2 . This production stops if and only if
either concentration of H2 is zero (c1 = 0) or concentration of O2 is zero (c2 = 0).
This means that the set of zeros of the irreversible reactions, c5,6,7 = 0 (c ≥ 0), is not invariant with
respect to the kinetics of the reversible reactions. This means that from an initial conditions on this set the
kinetic trajectory will leave it unless, in addition, c3 = c8 = 0 and either c1 = 0 or c2 = 0.
The reactions of all irreversible reactions should tend to zero due to Proposition 2. Therefore, the kinetic
trajectory should approach the union of two planes, c1,3,5,6,7,8 = 0 and c2,3,5,6,7,8 = 0 (under condition
c ≥ 0). These planes are two-dimensional and the position of the state there is completely defined by the
atomic balances.
If the concentration vector belongs to the first plane, then all the atoms are collected in O2 and H2 O. It
is possible if and only if bO ≥ 21 bH . In this case, c4 = 21 bH and c2 = 12 (bO − 21 bH ).
If the concentration vector belongs to the second plane, then all the atoms are collected in H2 and H2 O.
It is possible if and only if bO ≤ 12 bH . In this case, c4 = bO and c1 = 21 (bH − 2bO ).
Let us reproduce this reasoning formally using the general formalism of Sec. 3.3. Formula 20 gives for
S({5, 6, 7})
(c5 = 0)∧ ∧r,γr5 >0 ∨j,αrj >0 (cj = 0) ∧ ∧r,γr5 <0 ∨j,βrj >0 (cj = 0)
∧(c6 = 0)∧ ∧r,γr6 >0 ∨j,αrj >0 (cj = 0) ∧ ∧r,γr6 <0 ∨j,βrj >0 (cj = 0)
(22)
∧(c7 = 0)∧ ∧r,γr7 >0 ∨j,αrj >0 (cj = 0) ∧ ∧r,γr7 <0 ∨j,βrj >0 (cj = 0) .
Vectors γr that participate in this formula are the stoichiometric vectors of reversible reactions (r =
1, 2, 3, 4, 5, 9, 18, 19, 20). From the Table 1 we find that γr5 > 0 for r = 2, 3, 4, γr5 < 0 for r = 5, 19,
γr6 > 0 for r = 9, γr6 < 0 for r = 3, 4, γr7 > 0 for r = 5, 19, 20, and γr7 6< 0 for all r. Formula (22)
19
transforms into
(c5 = 0) ∧ ((c1 = 0)∨(c3 = 0)) ∧ ((c3 = 0)∨(c6 = 0)) ∧ ((c1 = 0)∨(c6 = 0)) ∧ (c7 = 0)
∧((c1 = 0)∨(c7 = 0)) ∧ (c6 = 0) ∧ (c3 = 0) ∧ ((c2 = 0)∨(c5 = 0)) ∧ ((c3 = 0)∨(c5 = 0))
∧(c7 = 0) ∧ ((c2 = 0)∨(c5 = 0)) ∧ ((c5 = 0)∨(c8 = 0)) ∧ ((c3 = 0)∨(c8 = 0)) .
After simple transformations it becomes
(c3 = 0) ∧ (c5 = 0) ∧ (c6 = 0) ∧ (c7 = 0) .
(23)
Therefore, S({5, 6, 7}) = {3, 5, 6, 7}. To iterate, we have to compute S({3, 5, 6, 7}). For this calculation, we
have to add one more line to formula (22), namely,
∧(c3 = 0) ∧ ∧r,γr3 >0 ∨j,αrj >0 (cj = 0) ∧ ∧r,γr3 <0 ∨j,βrj >0 (cj = 0) .
Let us take into account that γr3 > 0 for r = 1, 4, 18 and γr3 < 0 for r = 2, 3, 9, 20, and rewrite this formula
in the more explicit form
(c3 = 0) ∧ ((c1 = 0) ∨ (c2 = 0)) ∧ ((c1 = 0) ∨ (c6 = 0)) ∧ (c8 = 0)
∧((c4 = 0) ∨ (c5 = 0)) ∧ ((c2 = 0) ∨ (c5 = 0)) ∧ ((c4 = 0) ∨ (c6 = 0)) ∧ (c7 = 0) .
Let us take the conjunction of this formula with (22) taken in the simplified equivalent form (23) and
transform the result to the disjunctive form. We get
[(c3 = 0) ∧ (c5 = 0) ∧ (c6 = 0) ∧ (c7 = 0) ∧ (c8 = 0) ∧ (c1 = 0)]
∨[(c3 = 0) ∧ (c5 = 0) ∧ (c6 = 0) ∧ (c7 = 0) ∧ (c8 = 0) ∧ (c2 = 0))] .
(24)
This means that S2 ({5, 6, 7}) = S({3, 5, 6, 7}) = {{1, 3, 5, 6, 7, 8}, {2, 3, 5, 6, 7, 8}}. The further calculations
show that the next iteration does not change the result. Therefore, all the ω-limit points belong to two
faces, F{1,3,5,6,7,8} and F{2,3,5,6,7,8} . The result is the same as for the previous discussion. The detailed
formalization becomes crucial for more complex systems and for software development.
Let us find the vector of exponents δ = (δi ) (i = 1, . . . , 8) from the Table 2. After all the eliminations,
the corresponding linear functional δ̂ is just a value of the 7th coordinate: δ̂(x) = x7 . Its values are negative
(−2) for all irreversible reactions and zero for all reversible reactions (see the last column of the Table 2).
The conditions (δ, γ) = 0 for the reversible reactions and (δ, γ) < 0 for all irreversible reactions do not
define the unique vector: if δ satisfies these conditions then its linear combination with the vectors of atomic
balances also satisfy them. Such a combination is a vector
λδ + λH (2, 0, 1, 2, 1, 0, 1, 2) + λO (0, 2, 1, 1, 0, 1, 2, 2)
(25)
under condition λ > 0. This transformation of δ does not change the signs of δ̂ on the stoichiometric vectors
because of atomic balances.
In our case the only coordinate remains not eliminated, x7 (the bottom part of the last column of the
Table 2). If, for some reaction mechanism and selected sets of reversible and irreversible reaction, there
remain several (q) coordinates, then it is necessary to find q corresponding functionals δ̂ and the space
of possible vectors of exponents is (q + j)-dimensional. Here, j is the number of the independent linear
conservation laws for the whole system, j = n − rank{γr }, n is the number of the components, {γr } includes
all the stoichiometric vectors for reversible and irreversible reactions.
To find δ, we apply the elimination procedures from the Table 2 to an arbitrary vector y = (yi ) (i =
1, . . . , 8):
(y1 , y2 , y3 , y4 , y5 , y6 , y7 , y8 ) 7→ (y1 , y2 , 0, y4 , y5 , y6 , y7 , y8 )
7 (y1 , y2 , 0, y4 , y5 , y6 , y7 , 0) 7→ (y1 , y2 , 0, 0, y5 − y4 , y6 , y7 , 0)
→
7→(y1 , y2 , 0, 0, 0, y6 + y5 − y4 , y7 , 0) 7→ (y1 , y2 , 0, 0, 0, 0, y7 + y6 + y5 − y4 , 0) .
20
(26)
This sequence of transformations gives us the linear functional
δ̂(y) = y7 + y6 + y5 − y4 .
The corresponding vector of exponents (0, 0, 0, −1, 1, 1, 1, 0) should be corrected because its coordinates
cannot be negative. Let us apply (25) with λ = 2 (for convenience). The coordinates of this combination
are non-negative if and only if λH ≥ 0, λO ≥ 0 and 2λH + λO − 2 ≥ 0. The solutions of these linear
inequality on the (λH , λO ) plane is a convex combination of the extreme points (corners) (1, 0) and (0, 2)
plus any non-negative 2D vector: (λH , λO ) = ς(1, 0) + (1 − ς)(0, 2) + (ϑ1 , ϑ2 ), ϑ1,2 ≥ 0 and 1 ≥ ς ≥ 0. The
corresponding vectors of exponents are
(0, 0, 0, −2, 2, 2, 2, 0) + (ς + ϑ1 )(2, 0, 1, 2, 1, 0, 1, 2) + (1 − ς + ϑ2 )(0, 4, 2, 2, 0, 2, 4, 4) .
At least one of the exponents should be zero. There are only three possibilities, δ1 , δ2 or δ4 . For all other
i, δi > 0 if ϑ1,2 ≥ 0 and 1 ≥ ς ≥ 0.
To provide any necessary atomic balance in the limit ε → 0 it is necessary that two of δi are zeros. If
bO ≤ 12 bH , then δ1 = δ4 = 0. This means that ϑ1,2 = 0, ς = 0 and δ = (0, 4, 2, 0, 2, 4, 6, 4). It is convenient
to divide this δ by 2 and write
δ = (0, 2, 2, 0, 2, 2, 3, 2) .
For these exponents, the equilibrium concentrations tend to 0 with the small parameter ε → 0 (ε > 0) as
eq
eq
eq
eq
eq
eq
eq
2
2
ceq
H2 = c1 = const, cO2 = c2 ∼ ε , cOH = c3 ∼ ε , cH2 O = c4 = const,
(27)
eq
eq
eq
eq
eq
eq
eq
2
2
3
2
ceq
H = c5 ∼ ε , cO = c6 ∼ ε , cHO2 = c7 ∼ ε , cH2 O2 = c6 ∼ ε .
If bO ≥ 12 bH , then δ2 = δ4 = 0. This means that ϑ1,2 = 0, ς = 1 and
δ = (2, 0, 1, 0, 3, 2, 3, 2) .
For these exponents, the equilibrium concentrations tend to 0 with the small parameter ε → 0 (ε > 0) as
eq
eq
eq
eq
eq
eq
eq
2
ceq
H2 = c1 ∼ ε , cO2 = c2 = const, cOH = c3 ∼ ε, cH2 O = c4 = const,
(28)
eq
eq
eq
eq
eq
eq
eq
2
3
3
2
ceq
H = c5 ∼ ε , cO = c6 ∼ ε , cHO2 = c7 ∼ ε , cH2 O2 = c6 ∼ ε .
P
The linear combination i δi Ni decreases in time due to kinetic equations. This is true for any vector
of exponents presented by a linear combination (25) (λ 6= 0) of the initial vector (0, 0, 0, −1, 1, 1, 1, 0) with
the vectors of the atomic balances. At the same time, any of these combinations give an additional linear
conservation law for the system of reversible reactions.
Below are several versions of this function:
• The initial version, δ̂, obtained from the Table 2 is (δ, N ) = −NH2 0 + NH + NO + NH O2 ;
• Vector of exponents, calibrated by adding of the atomic balances (25) to meet the atomic balance
conditions for bO ≤ 21 bH in the limit ε → 0 is (δ, N ) = 2NO2 + 2NOH + 2NH + 2NO + 3NHO2 + 2NH2 O2 ;
• Vector of exponents, calibrated to meet the atomic balance conditions for bO ≥
2NH2 + NOH + 3NH + 2NO + 3NHO2 + 2NH2 O2 .
1
2 bH
is, (δ, N ) =
All these forms differs by the combinations of the atomic balances (25) and are, in this sense, equivalent.
5. Conclusion
The general principle of detailed balance was formulated in 1925 as follows [15]: “Corresponding to every
individual process there is a reverse process, and in a state of equilibrium the average rate of every process
is equal to the average rate of its reverse process.” Rigorously speaking, the chemical reactions have to
21
be considered as reversible ones, and every step of the complex reaction consists of two reactions, forward
and reverse (backward) one. However, in reality the rates of some forward or reverse reactions may be
negligible. Typically, the complex combustion reactions, in particular, reactions of hydrocarbon oxidation
or hydrogen combustion, include both reversible and irreversible steps. It is a case in catalytic reactions as
well. In particular, many enzyme reactions are “partially irreversible”. Although many catalytic reactions
are globally irreversible, they always include some reversible steps, in particular steps of adsorption of gases.
Many enzyme reactions are also “partially irreversible”.
This work aims to solve the problem of the partially irreversible limit in chemical thermodynamics when
some reactions become irreversible whereas some other reactions remain reversible. The main results in this
direction are
1. Description of the multiscale limit of a system reversible reactions when some of equilibrium concentrations tend to zero (Sec. 2.2).
2. Extended principle of detailed balance for the systems with some irreversible reactions (Theorem 1).
3. The linear functional Gδ that decreases in time on solutions of the kinetic equations under the extended
detailed balance conditions (Proposition 2 and Eq. (11)).
4. The entropy production (or free energy dissipation) formulas for the reversible part of the reaction
mechanism under the extended detailed balance conditions (Eqs. (13), (15)).
5. Description of the faces of the positive orthant which include the ω-limit points in their relative interior
and, therefore, description of limiting behavior in time (Theorem 2).
Did we solve the main problem and create the thermodynamic of the systems with some irreversible
reaction? The answer is: we solved this problem partially. We described the limit behavior but we did not
find the global Lyapunov function that captures relaxation of both reversible and irreversible parts of the
system. The good candidate is a linear combination of the relevant classical thermodynamic potential and
Gδ but we did not find the coefficients. In that sense, the problem of the limit thermodynamics remains
open.
Nevertheless, one problem is solved ultimately and completely: How to throw away some reverse reactions without violation of thermodynamics and microscopic reversibility? The answer is: the convex hull
of the stoichiometric vectors of the irreversible reactions should not intersect with the linear span of the
stoichiometric vectors of the reversible reactions and the reaction rate constants of the remained reversible
reactions should satisfy the Wegscheider identities (8).
The solution of this theoretical problem is important for the modeling of the chemical reaction networks.
This is because some of reactions are practically irreversible. Removal of some reverse reaction from the
reaction mechanism cannot be done independently of the whole structure of the reaction network; the whole
reaction mechanism should be used in the decision making.
If the irreversible reactions are introduced correctly then we also know that the closed system with this
reaction mechanism goes to an equilibrium state. At this equilibrium, all the reaction rates are zero: the
irreversible reaction rates vanish and the rates of the reversible reactions satisfy the principle of detailed
balance. The limit equilibria are situated on the faces of the positive orthant of concentrations and these
faces are described in the paper.
The oscillatory or chaotic attractors are impossible in closed systems which satisfy the extended principle
of detailed balance. This general statement can be considered as a simple consequence of thermodynamics.
It can be easily proved if the thermodynamic Lyapunov functions (potentials) are given. However, the
thermodynamic potentials have no limits for the systems with some irreversible reactions and we do not
know a priori any general theorem that prohibits bifurcations at the zero values of some reaction rate
constants. In this paper we proved, in particular, that the emergency of nontrivial attractors in systems
with some irreversible reactions is impossible if they are the limits of the reversible systems which satisfy the
principle of detailed balance. In this sense, the thermodynamic behavior is proven for the systems with some
irreversible reactions under the extended detailed balance conditions. Nevertheless, the general problem of
the thermodynamic potentials in this limit remains open.
22
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
D. Bertsimas, J.N. Tsitsiklis, Introduction to Linear Optimization, Athena Scientific, Cambridge, MA, USA, 1997.
L. Boltzmann, Lectures on gas theory, Univ. of California Press, Berkeley, CA, USA, 1964.
B.R. Clarke, Challenging Logic Puzzles, Sterling Publishing Co., New York, NY, USA, 2003.
D. Colquhoun, K.A. Dowsland, M. Beato, A.J.R. Plested, How to Impose Microscopic Reversibility in Complex Reaction
Mechanisms, Biophysical Journal 86 (6) (2004) 3510–3518.
T. de Donder, P. van Rysselberghe, Thermodynamic Theory of Affinity. A Book of Principles, Stanford Univ. Press, 1936.
M. Feinberg. On chemical kinetics of a certain class. Arch. Rat. Mechan. Anal. 46 (1972) 1–41.
V. Giovangigli, L. Matuszewski, Supercritical fluid thermodynamics from equations of state, Physica D 241 (6) (2012)
649–670.
A.N. Gorban, Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis, Nauka, Novosibirsk, 1984.
A.N. Gorban, Singularities of transition processes in dynamical systems: Qualitative theory of critical delays, Electron. J.
Diff. Eqns., Monograph 05, 2004. http://ejde.math.txstate.edu/Monographs/05/abstr.html
A.N. Gorban, E.M. Mirkes, A.N. Bocharov, V.I. Bykov, Thermodynamic consistency of kinetic data, Combustion, Explosion, and Shock Waves 25 (5) (1989) 593–600.
A.N. Gorban, M. Shahzad, The Michaelis–Menten–Stueckelberg Theorem, Entropy 13 (5) (2011) 966–1019;
arXiv:1008.3296 [physics.chem-ph].
A.N. Gorban, G.S. Yablonskii, Extended detailed balance for systems with irreversible reactions, Chem. Eng. Sci. 66
(2011) 5388–5399; arXiv:1101.5280 [cond-mat.stat-mech].
M. Grmela, Multiscale equilibrium and nonequilibrium thermodynamics in chemical engineering, Adv. Chem. Eng. 39
(2010) 75–128.
M. Grmela, Fluctuations in extended mass-action-law dynamics, Physica D 241 (10) (2012) 976–986.
G.N. Lewis, A new principle of equilibrium, P.N.A.S. USA 11 (3) (1925), 179–183.
J. Li, Zh. Zhao, A. Kazakov, and F.L. Dryer, An Updated Comprehensive Kinetic Model of Hydrogen Combustion, Int.
J. Chem. Kinet. 36 (2004) 566–575.
I. Prigogine, R. Defay, Chemical Thermodynamics, Longmans and Green, New York, NY, USA, 1962.
P. Saxena and F. A. Williams. Testing a small detailed chemical-kinetic mechanism for the combustion of hydrogen and
carbon monoxide. Combustion and Flame 145 (2006) 316–23.
N.Z. Shapiro, L.S. Shapley, Mass action law and the Gibbs free energy function, SIAM J. Appl. Math. 16 (1965) 353–375.
S. Schuster, R. Schuster, A generalization of Wegscheider’s condition. Implications for properties of steady states and for
quasi-steady-state approximation, J. Math. Chem. 3 (1) (1989) 25–42.
D.G. Vlachos, Reduction of detailed kinetic mechanisms for ignition and extinction of premixed hydrogen/air flames,
Chem. Eng. Sci. 51 (16) (1996) 3979–3993.
A.I. Volpert, S.I. Khudyaev, Analysis in classes of discontinuous functions and equations of mathematical physics, Nijoff.
Dordrecht, The Netherlands, 1985. (Translation from the 1st Russian ed., Moscow, Nauka publ., 1975.)
R. Wegscheider, Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik
homogener Systeme, Monatshefte für Chemie / Chemical Monthly 32 (8) (1901) 849–906.
G.S. Yablonskii, V.I. Bykov, A.N. Gorban, V.I. Elokhin, Kinetic Models of Catalytic Reactions, Elsevier, Amsterdam,
The Netherlands, 1991.
J. Yang, W.J. Bruno, W.S. Hlavacek, J. Pearson, On imposing detailed balance in complex reaction mechanisms. Biophys.
J. 91 (2006) 1136–1141.
Y.B. Zeldovich, Proof of the uniqueness of the solution of the equations of the law of mass action. In: Selected Works
of Yakov Borisovich Zeldovich; Volume 1, Ed. by J.P. Ostriker; Princeton Univ. Press, Princeton, NJ, USA, 1996; pp.
144–148.
23